341 lines
9.8 KiB
Python
341 lines
9.8 KiB
Python
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"""
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This module has all the classes and functions related to waves in optics.
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**Contains**
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* TWave
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"""
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__all__ = ['TWave']
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from sympy.core.basic import Basic
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from sympy.core.expr import Expr
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from sympy.core.function import Derivative, Function
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from sympy.core.numbers import (Number, pi, I)
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from sympy.core.singleton import S
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from sympy.core.symbol import (Symbol, symbols)
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from sympy.core.sympify import _sympify, sympify
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from sympy.functions.elementary.exponential import exp
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from sympy.functions.elementary.miscellaneous import sqrt
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from sympy.functions.elementary.trigonometric import (atan2, cos, sin)
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from sympy.physics.units import speed_of_light, meter, second
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c = speed_of_light.convert_to(meter/second)
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class TWave(Expr):
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r"""
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This is a simple transverse sine wave travelling in a one-dimensional space.
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Basic properties are required at the time of creation of the object,
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but they can be changed later with respective methods provided.
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Explanation
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===========
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It is represented as :math:`A \times cos(k*x - \omega \times t + \phi )`,
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where :math:`A` is the amplitude, :math:`\omega` is the angular frequency,
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:math:`k` is the wavenumber (spatial frequency), :math:`x` is a spatial variable
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to represent the position on the dimension on which the wave propagates,
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and :math:`\phi` is the phase angle of the wave.
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Arguments
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=========
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amplitude : Sympifyable
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Amplitude of the wave.
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frequency : Sympifyable
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Frequency of the wave.
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phase : Sympifyable
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Phase angle of the wave.
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time_period : Sympifyable
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Time period of the wave.
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n : Sympifyable
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Refractive index of the medium.
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Raises
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=======
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ValueError : When neither frequency nor time period is provided
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or they are not consistent.
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TypeError : When anything other than TWave objects is added.
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Examples
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========
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>>> from sympy import symbols
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>>> from sympy.physics.optics import TWave
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>>> A1, phi1, A2, phi2, f = symbols('A1, phi1, A2, phi2, f')
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>>> w1 = TWave(A1, f, phi1)
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>>> w2 = TWave(A2, f, phi2)
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>>> w3 = w1 + w2 # Superposition of two waves
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>>> w3
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TWave(sqrt(A1**2 + 2*A1*A2*cos(phi1 - phi2) + A2**2), f,
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atan2(A1*sin(phi1) + A2*sin(phi2), A1*cos(phi1) + A2*cos(phi2)), 1/f, n)
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>>> w3.amplitude
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sqrt(A1**2 + 2*A1*A2*cos(phi1 - phi2) + A2**2)
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>>> w3.phase
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atan2(A1*sin(phi1) + A2*sin(phi2), A1*cos(phi1) + A2*cos(phi2))
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>>> w3.speed
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299792458*meter/(second*n)
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>>> w3.angular_velocity
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2*pi*f
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"""
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def __new__(
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cls,
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amplitude,
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frequency=None,
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phase=S.Zero,
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time_period=None,
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n=Symbol('n')):
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if time_period is not None:
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time_period = _sympify(time_period)
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_frequency = S.One/time_period
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if frequency is not None:
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frequency = _sympify(frequency)
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_time_period = S.One/frequency
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if time_period is not None:
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if frequency != S.One/time_period:
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raise ValueError("frequency and time_period should be consistent.")
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if frequency is None and time_period is None:
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raise ValueError("Either frequency or time period is needed.")
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if frequency is None:
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frequency = _frequency
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if time_period is None:
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time_period = _time_period
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amplitude = _sympify(amplitude)
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phase = _sympify(phase)
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n = sympify(n)
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obj = Basic.__new__(cls, amplitude, frequency, phase, time_period, n)
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return obj
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@property
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def amplitude(self):
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"""
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Returns the amplitude of the wave.
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Examples
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========
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>>> from sympy import symbols
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>>> from sympy.physics.optics import TWave
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>>> A, phi, f = symbols('A, phi, f')
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>>> w = TWave(A, f, phi)
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>>> w.amplitude
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A
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"""
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return self.args[0]
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@property
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def frequency(self):
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"""
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Returns the frequency of the wave,
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in cycles per second.
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Examples
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========
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>>> from sympy import symbols
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>>> from sympy.physics.optics import TWave
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>>> A, phi, f = symbols('A, phi, f')
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>>> w = TWave(A, f, phi)
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>>> w.frequency
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f
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"""
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return self.args[1]
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@property
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def phase(self):
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"""
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Returns the phase angle of the wave,
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in radians.
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Examples
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========
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>>> from sympy import symbols
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>>> from sympy.physics.optics import TWave
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>>> A, phi, f = symbols('A, phi, f')
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>>> w = TWave(A, f, phi)
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>>> w.phase
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phi
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"""
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return self.args[2]
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@property
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def time_period(self):
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"""
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Returns the temporal period of the wave,
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in seconds per cycle.
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Examples
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========
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>>> from sympy import symbols
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>>> from sympy.physics.optics import TWave
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>>> A, phi, f = symbols('A, phi, f')
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>>> w = TWave(A, f, phi)
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>>> w.time_period
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1/f
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"""
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return self.args[3]
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@property
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def n(self):
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"""
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Returns the refractive index of the medium
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"""
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return self.args[4]
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@property
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def wavelength(self):
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"""
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Returns the wavelength (spatial period) of the wave,
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in meters per cycle.
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It depends on the medium of the wave.
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Examples
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========
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>>> from sympy import symbols
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>>> from sympy.physics.optics import TWave
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>>> A, phi, f = symbols('A, phi, f')
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>>> w = TWave(A, f, phi)
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>>> w.wavelength
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299792458*meter/(second*f*n)
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"""
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return c/(self.frequency*self.n)
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@property
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def speed(self):
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"""
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Returns the propagation speed of the wave,
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in meters per second.
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It is dependent on the propagation medium.
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Examples
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========
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>>> from sympy import symbols
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>>> from sympy.physics.optics import TWave
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>>> A, phi, f = symbols('A, phi, f')
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>>> w = TWave(A, f, phi)
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>>> w.speed
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299792458*meter/(second*n)
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"""
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return self.wavelength*self.frequency
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@property
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def angular_velocity(self):
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"""
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Returns the angular velocity of the wave,
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in radians per second.
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Examples
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========
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>>> from sympy import symbols
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>>> from sympy.physics.optics import TWave
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>>> A, phi, f = symbols('A, phi, f')
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>>> w = TWave(A, f, phi)
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>>> w.angular_velocity
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2*pi*f
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"""
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return 2*pi*self.frequency
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@property
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def wavenumber(self):
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"""
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Returns the wavenumber of the wave,
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in radians per meter.
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Examples
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========
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>>> from sympy import symbols
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>>> from sympy.physics.optics import TWave
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>>> A, phi, f = symbols('A, phi, f')
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>>> w = TWave(A, f, phi)
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>>> w.wavenumber
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pi*second*f*n/(149896229*meter)
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"""
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return 2*pi/self.wavelength
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def __str__(self):
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"""String representation of a TWave."""
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from sympy.printing import sstr
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return type(self).__name__ + sstr(self.args)
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__repr__ = __str__
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def __add__(self, other):
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"""
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Addition of two waves will result in their superposition.
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The type of interference will depend on their phase angles.
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"""
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if isinstance(other, TWave):
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if self.frequency == other.frequency and self.wavelength == other.wavelength:
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return TWave(sqrt(self.amplitude**2 + other.amplitude**2 + 2 *
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self.amplitude*other.amplitude*cos(
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self.phase - other.phase)),
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self.frequency,
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atan2(self.amplitude*sin(self.phase)
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+ other.amplitude*sin(other.phase),
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self.amplitude*cos(self.phase)
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+ other.amplitude*cos(other.phase))
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)
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else:
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raise NotImplementedError("Interference of waves with different frequencies"
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" has not been implemented.")
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else:
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raise TypeError(type(other).__name__ + " and TWave objects cannot be added.")
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def __mul__(self, other):
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"""
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Multiplying a wave by a scalar rescales the amplitude of the wave.
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"""
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other = sympify(other)
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if isinstance(other, Number):
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return TWave(self.amplitude*other, *self.args[1:])
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else:
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raise TypeError(type(other).__name__ + " and TWave objects cannot be multiplied.")
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def __sub__(self, other):
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return self.__add__(-1*other)
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def __neg__(self):
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return self.__mul__(-1)
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def __radd__(self, other):
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return self.__add__(other)
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def __rmul__(self, other):
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return self.__mul__(other)
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def __rsub__(self, other):
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return (-self).__radd__(other)
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def _eval_rewrite_as_sin(self, *args, **kwargs):
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return self.amplitude*sin(self.wavenumber*Symbol('x')
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- self.angular_velocity*Symbol('t') + self.phase + pi/2, evaluate=False)
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def _eval_rewrite_as_cos(self, *args, **kwargs):
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return self.amplitude*cos(self.wavenumber*Symbol('x')
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- self.angular_velocity*Symbol('t') + self.phase)
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def _eval_rewrite_as_pde(self, *args, **kwargs):
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mu, epsilon, x, t = symbols('mu, epsilon, x, t')
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E = Function('E')
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return Derivative(E(x, t), x, 2) + mu*epsilon*Derivative(E(x, t), t, 2)
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def _eval_rewrite_as_exp(self, *args, **kwargs):
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return self.amplitude*exp(I*(self.wavenumber*Symbol('x')
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- self.angular_velocity*Symbol('t') + self.phase))
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